Since the TFM from (REF ) is stable, while (REF )-() and (REF )-() are stabilizable and detectable, we apply the negative unity feedback versions of Lemmas 5.2 and 5.3 from [1]} to the feedback loop from Fig. REF  (directly in the continuous-time case and in adapted form for discrete-time), to get that \(\Lambda _u(A_{CL})=\lbrace \emptyset \rbrace \) . This is equivalent, by Definition 5.2 in [1]} (with the appropriate alteration for the discrete-time case), to the desired result, i.e., any finite initial conditions of both the plant's and the distributed controller's state vectors in Fig. REF  or Fig. REF  are driven asymptotically to the zero vector. \(\Box \)
